ISSN 2590-9770 The Art of Discrete and Applied Mathematics 3 (2020) #P1.02 https://doi.org/10.26493/2590-9770.1291.c54 (Also available at http://adam-journal.eu) On strongly sequenceable abelian groups Brian Alspach , Georgina Liversidge School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia Received 4 February 2019, accepted 7 August 2019, published online 22 July 2020 Abstract A group is strongly sequenceable if every connected Cayley digraph on the group ad- mits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the problem of whether finite abelian groups are strongly sequenceable. A method based on posets is used to show that if the connection set for a Cayley digraph on an abelian group has cardinality at most nine, then the digraph admits either an orthogonal directed path or an orthogonal directed cycle. Keywords: Strongly sequenceable, abelian group, diffuse poset, sequenceable poset. Math. Subj. Class. (2020): 05C25 1 Introduction The Cayley digraph −−→ Cay(G; S) on the group G has the elements of G for the vertex set and an arc (g,h) from g to h whenever h = gs for some s ∈ S, where S ⊂ G and 1 ∈ S. The set S is called the connection set. It is easy to see that left-multiplication by any element of G is an automorphism of −−→ Cay(G; S) which implies that the automorphism group of −−→ Cay(G; S) contains the left-regular representation of G. A given s ∈ S generates a spanning digraph of −−→ Cay(G; S) composed of vertex-disjoint directed cycles of length |s|, where |s| denotes the order of s. We call this subdigraph a (1, 1)-directed factor because the in-valency and out-valency at each vertex is 1. Hence, there is a natural factorization of −−→ Cay(G; S) into |S| arc-disjoint (1, 1)-directed factors. This is the Cayley factorization of −−→ Cay(G; S) and is denoted F (G; S). Let −−→ Cay(G; S) be a Cayley digraph on a group G. A subdigraph − → Y of −−→ Cay(G; S) of size |S| (the size is the number of arcs in − → Y ), is orthogonal to F (G; S) if − → Y has one arc E-mail addresses: brian.alspach@newcastle.edu.au (Brian Alspach), gliv560@aucklanduni.ac.nz (Georgina Liversidge) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/